LEDs and resistors - beginner question

All of the LEDs I've been connecting to my Uno have their own resistors like this:

What would happen if all of the LEDs were connected to a common wire (on the negative side, not the Uno pin side) that had a single 330 ohm resistor going to ground?

Thanks,
Jake

LED's would light up with different brightness.

Well you should avoid that because it is not a good practice.
The output pin can just delivery 40 mA so if you exceed that you will damage the output.In theroy if you calculate a resitor for not go to the max, you can do it but the brightness on led will be very low.
If you want to drive all LEDs on the same pin you must use a transistor for get more power to then

Say you had 1 LED, Vf of 2V, and 330 ohm resistor.
Uno pin high.
Vresistor = 3V (5V - 2V). Current = 3V/330 ohm = 9mA.
Now a 2nd LED turns on. You effectively have two 3V sources connected to 330 ohm resistor. Same 9mA flows thru.
Each LED then gets ~4.5mA.
Add a 3rd - each gets ~3mA.
4th - each gets ~2.5mA.
So you end up with the varying brightness Magician refers to.

Thanks for the detailed explanation Robert. That makes sense.

Its not recommended but will sort of work - the LEDs must be the same kind (different colours are different voltages) and at the same temperature. Because of manufacturing variablility the LEDs will be different brightnesses even if the same colour, but the eye won't notice (measuring the current each one takes will show this difference though.

If all from the same batch then more likely to be matched - this is done all the time in battery-powered xmas lights for instance.

There is another issue with connecting in parallel if they are high-power LEDs - the hotter the LED the lower the voltage it needs - this means that as one LED get hotter it steals more current from the others, which makes it dissipate more power and get hotter, a vicious circle...

Overall its best to have separate resistors to keep the individual currents the same.